Optimal. Leaf size=102 \[ -\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \text {ArcSin}(c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 4723,
331, 335, 227} \begin {gather*} -\frac {2 (a+b \text {ArcSin}(c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}}-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 331
Rule 335
Rule 4723
Rule 4889
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{5/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{15 d e^3}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{15 d e^4}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.03, size = 59, normalized size = 0.58 \begin {gather*} \frac {-6 (a+b \text {ArcSin}(c+d x))-4 b (c+d x) \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};(c+d x)^2\right )}{15 d e (e (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 169, normalized size = 1.66
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(169\) |
default | \(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.79, size = 165, normalized size = 1.62 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, b d^{2} \arcsin \left (d x + c\right ) + 3 \, a d^{2} + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d x + c} e^{\frac {1}{2}} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-4\right )}}{15 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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